Stat 371 Spring 2011 Assignment 2
Solutions
1.
(Some fun with covariances) Appendix 2 in the course notes deals with
expectation, variance and covariance for vectors of random variables. This
question will give you some practice playing with the formulas.
a)
If
U
and
V
are two vectors of random variables of length
m
and
n
respectively, and
is an
[
,
]
Cov U V
m
n
×
matrix with
element
, prove that
th
ij
[
,
]
j
i
U V
j
Cov
[
,
]
[(
[
])(
[
]) ]
T
Cov U V
E U
E U
V
E V
=
−
−
We have
and the matrix
is
m
with
entry (
and so the result follows.
[
,
]
[(
[
])(
[
])]
i
j
i
i
j
j
Cov U V
E U
E U
V
E V
=
−
−
(
[
])(
[
])
T
U
E U
V
E V
−
−
n
×
th
ij
[
])(
[
])
i
i
j
U
E U
V
E V
−
−
b)
Show that
[
,
]
[
,
]
Cov AU V
ACov U V
=
and
[
,
]
[
,
]
T
Cov U BV
Cov U V B
=
Using the results in a), we have
[
,
]
[(
[
])(
[
])
[
(
[
])(
[
]) ]
[(
[
])(
[
]) ]
T
T
T
Cov AU V
E
AU
E AU
V
E V
E A U
AE U
V
E V
AE U
E U
V
E V
=
−
−
=
−
−
=
−
−
]
and
[
,
]
[(
[
])(
[
]) ]
[(
[
])(
[
]) ]
[(
[
]){ (
[
])} ]
[(
[
])(
[
])
]
T
T
T
T
T
Cov U BV
E U
E U
BV
E BV
E U
E U
BV
E BV
E U
E U
B V
E V
E U
E U
V
E V
B
=
−
−
=
−
−
=
−
−
=
−
−
c)
For the regression model
Y
X
R
β
=
+
where
2
~
(0,
)
R
N
σ
I
0
, show that
and
[
, ]
Cov X
r
β
=
±
±
[ , ]
0
Cov Y r
≠
±
2
[
, ]
[
,(
)
]
[ ,
](
)
(
)
0
T
Cov X
r
Cov HY
I
H Y
HCov Y Y
I
H
H
I I
H
β
σ
=
−
=
−
=
−
=
±
±
since (
)
I
H
−
is symmetric.
2
[ , ]
[ ,(
)
]
[ ,
](
)
(
)
T
Cov Y r
Cov Y
I
H Y
Cov Y Y
I
H
I
H
σ
=
−
=
−
=
−
±
which is not a matrix of 0’s.

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